Spie Press Book

Here for the first time is an integrated mathematical view of the physics and numerical modeling of optical projection lithography that efficiently covers the full spectrum of the important concepts. Alfred Wong offers rigorous underpinning, clarity in systematic formulation, physical insight into emerging ideas, as well as a system-level view of the parameter tolerances required in manufacturing. Readers with a good working knowledge of calculus can follow the step-by-step development, and technologists can gather general concepts and the key equations that result. Even the casual reader will gain a perspective on the key concepts, which will likely help facilitate dialog among technologists.

Book Details

Date Published: 9 March 2005Pages: 276ISBN: 9780819458292Volume: TT66

Table of Contents

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Foreword xiii

Preface xv

List of Symbols xvii

1 Basic Electromagnetism 1

1.1 Maxwell's equations 1

1.2 Electromagnetic energy 3

1.3 The wave equation 5

1.4 Plane waves 6

1.5 Spherical waves 9

1.6 Harmonic waves 9

1.7 Quasi-monochromatic light 14

2 Elements of Geometrical Optics 19

2.1 The eikonal equation 19

2.2 Light rays 20

2.3 Snell's law 23

2.4 Thin lens 24

2.5 Representation of an exposure system 26

3 Elements of Diffraction Theory 31

3.1 Qualitative consideration 31

3.2 Reciprocity 33

3.3 The Helmholtz-Kirchhoff theorem 34

3.4 Fresnel-Kirchhoff diffraction 35

3.5 The Rayleigh-Sommerfeld diffraction formula 39

3.6 Fraunhofer diffraction 41

3.7 Fraunhofer diffraction patterns 45

3.7.1 Rectangular pupil 45

3.7.2 Circular and annular pupils 46

4 Imaging of Extended Objects with Finite Sources 51

4.1 Coherent illumination 51

4.2 Obliquity factor 56

4.3 Spatial correlation of light 58

4.3.1 Mutual intensity and complex degree of coherence 58

4.3.2 Extended incoherent quasi-monochromatic source 61

4.3.3 Propagation of the mutual intensity 64

4.4 Kohler's illumination method 65

4.5 Partially coherent imaging 67

5 Resolution and Image Enhancement 75

5.1 Image intensity spectrum 76

5.2 Binary intensity objects under on-axis illumination 78

5.3 Off-axis illumination 83

5.4 Attenuated phase-shifting mask 85

5.5 Alternating phase-shifting mask 87

5.6 Minimum half-pitch 89

5.7 Minimum dimension 90

6 Oblique Rays 97

6.1 Polarization 97

6.2 Vector imaging 102

6.3 Wave propagation across a dielectric interface 107

6.3.1 The laws of reflection and refraction 108

6.3.2 Reflected and transmitted wave amplitudes 109

6.3.3 Reflectivity and transmissivity 112

6.3.4 Polarization upon reflection and transmission 114

6.3.5 Total internal reflection 115

6.4 Stratified media 116

6.4.1 Basic equations 117

6.4.2 Characteristic matrix 119

6.4.3 Reflection and transmission 122

6.5 Intensity distribution in photoresist 124

6.6 Immersion imaging 126

6.7 Imaging with oblique rays 127

7 Aberrations 133

7.1 Diffraction of an aberrated wavefront 133

7.2 General properties of the aberration function 135

7.2.1 Displacement theorem 135

7.2.2 Intensity and average wavefront deformation 136

7.3 Zernike polynomials 137

7.4 Effects on imaging 140

7.5 Measurement 143

7.5.1 Interferometry 144

7.5.2 The extended Nijboer-Zernike approach 144

7.5.3 The Hartmann test 146

7.5.4 Aberration monitor patterns 147

8 Numerical Computation 151

8.1 Imaging equations 151

8.2 Transmission cross-coefficient integration 153

8.3 Source points integration 155

8.4 Coherent decomposition 157

8.5 Object spectrum 159

8.6 Remarks 162

9 Variabilities 165

9.1 Categorization 165

9.2 Proximity effect 167

9.3 Object variabilities (photomask errors) 170

9.3.1 Dimensional error 170

9.3.2 Phase and transmission errors 174

9.3.3 Edge roughness 175

9.4 Polarization effects 176

9.5 Illumination 177

9.6 Pupil 179

9.7 Focus 179

9.8 Dose 182

9.9 Flare 183

9.10 Remarks 186

A Birefringence 191

B Stationarity and Ergodicity 197

C Some Zernike Polynomials 199

D Simulator Accuracy Tests 205

D.1 Blank mask 205

D.2 Images of M = 1 systems 207

D.2.1 Chromium-on-glass mask under on-axis illumination 207

D.2.2 Dipole illumination of attenuated phase-shifting mask 208

D.2.3 Equal line-space on alternating phase-shifting mask 209

D.2.4 Periodic contacts on chromium-on-glass mask 209

D.3 Aberrations 210

D.4 Finite number of source points 210

D.4.1 Resist image with aerial coupling medium 210

D.4.2 Immersion imaging 212

E Select Refractive Indexes 215

F Assorted Theorems and Identities 217

Bibliography 219

Solutions to Exercises 231

Index 249

Foreword

Lithographers have pushed optical projection printing well beyond the imagined limits.
Even microwave engineers, quantum physicists, and photonic scientists would not have
expected the majority of integrated circuit chips in existence today to have feature
sizes below the fundamental half-wavelength limit of waveguide modes, eigenstates, and
stop bands. This has been accomplished by the introduction of innovations by many
technologists. These innovations include off-axis illumination, phase-shifting masks,
measurement of aberrations and flare, high-NA imaging, immersion, resist coatings, and
photomask precompensation for optical and even process effects. Equally as important
has been the ability to integrate these innovations as part of a complete system and
simultaneously optimize the interplay of their key parameters.

Here for the first time is an integrated mathematical view of the physics and
numerical modeling of projection printing that efficiently covers the full spectrum of
the important concepts. This book is far broader than the material found in any
optical text or reference book. Alfred Wong works from his firsthand involvement in
semiconductor manufacturing and his interest in theoretical concepts. His writing
is like the circuits he once designed in that it performs the desired function
quickly, without wasted energy, space or glitches. Alfred Wong pulls together the
diverse aspects in a common framework with a solid physical foundation. The framework
is used to give intuitive explanations, models for quantitative characterization,
tolerances for controlling variations and insight into simulation methodologies. The
broad scope includes many second-order effects that dominate production and design
for manufacturing strategies.

Specifically, the value of this book is that it systematically redevelops and extends
in one notation the rather challenging and extensive set of theoretical concepts used
in advanced projection printing systems and their numerical simulation. In a sense, it
is all of those one-assumption-at-a-time steps and messy equations that there is never
time nor space for in a conference paper. By starting from the basics of Maxwell's
equations, ray-tracing and diffraction, the important conceptual elements associated
with them are nicely summarized. The extension of the formulation of imaging to the
full optical system gives a particularly clear treatment of several advanced concepts.
These include the relationship of the illumination spectrum to the spatial degree of
coherence across the mask, the simplification of the formulation when the image only
depends on differences in distances, and when the diffraction efficiency of the mask
is independent of the angle of incidence.

Through a simplifying assumption about the mask spectrum, the formulation is extended
to give both new and previously known equations for image modulation, cut-off limits,
mask requirements, and unwanted side effects. Full formulations are made for the new
high-NA challenges of polarization, vector imaging, resist materials and immersion.
Both aberration effects on imaging and the advanced theoretical concepts for
monitoring them are considered. The parallel treatment of the Abbe, the Hopkins, and
the more advanced sum of coherent system approaches for image calculation clarifies
their methodologies and advantages. The identification and parameterization of some
eleven sources of variability in imaging is in itself a framework for characterizing
exposure tools.

Readers with a good working knowledge of calculus can follow through the step by step
development. A technologist may want to get the general idea of each concept and then
skip ahead to the key characterization equations that result. Even the casual reader
will gain a perspective on the key concepts and this will likely help facilitate
dialog among technologists much in the way that the k1 and k2 parameters of Burn Lin
have done.

This book is a tribute to the advanced concepts and innovation used in the field of
projection printing over the last 30 years. Alfred Wong offers rigorous underpinning,
clarity in systematic formulation, physical insight into emerging ideas, as well as a
system level view of the parameter tolerances required in manufacturing. This book is
very much the "Born and Wolf of Projection Printing" and Alfred is our seminal
chanticleer for the concepts that support this practice.

Andrew R. Neureuther

Preface

Optical projection lithography will remain the predominant microlithography technology
in the foreseeable future. With 193-nm radiation and an immersed numerical aperture of
1.38, the k1 factor of a 45-nm feature is 0.32. Fabrication at such low k1 factors
requires both image enhancement and tight control of process fluctuations.

A prerequisite to successful resolution improvement and variability control is an
understanding of optical imaging fundamentals. This book aims to explicate the
principles of image formation in projection microlithography, balancing intuitive
understanding with mathematical rigor such that the readers can both distill the
essence of the physics and form a firm foundation from which imaging techniques can be
analyzed and developed.

Chapter 1 derives the properties of light that are relevant for analysis of image
formulation in photolithography. From Maxwell's equations we deduce that light is a
transverse wave, with the electric and magnetic field vectors vibrating in a plane
that is normal to its direction of propagation. When light interacts with objects
whose physical dimensions are large compared with its wavelength, we can neglect the
field vectors under many circumstances, and approximate Maxwell's equations by laws
formulated in the language of geometry. This topic of geometrical optics is treated in
Chapter 2. To describe light transmission through apertures whose dimensions are
comparable to or smaller than the wavelength, however, we need to resort to
diffraction theory, a subject we discuss in Chapter 3.

Photomasks used in optical lithography require illumination by light sources that are
physically extended. Despite incoherence between source points making up the extended
source, vibrations at different object points are correlated due to diffraction of the
illumination optics. Chapter 4 develops the concept of spatial coherence and the
associated mutual intensity function that enable mathematical description of partially
coherent imaging scenarios. The resulting equations are used in Chapter 5 to examine
the theoretical and practical limits of the minimum dimension and the minimum
half-pitch.

Based on the foundation of the first five chapters, we further our development to
address topics that are becoming crucial as microlithographers push the limits of
optical imaging. The use of high-numerical-aperture lenses necessitates consideration
of the directional nature of light vibrations. Chapter 6 formulates the vector theory
of imaging that is applicable for immersion lithography in the presence of a
stratified wafer stack. Simultaneous with increasing numerical aperture are stringent
aberration requirements. The impact of lens aberrations is explored through
diffraction theory in Chapter 7.

Our abilities to harness the power of affordable computers to predict images of object
patterns, and to optimize the photomask and exposure configuration given a desired
image are becoming indispensable. Chapter 8 discusses common numerical approaches for
imaging simulation. Variability control is also integral for successful low-k1
lithography, as both layout shapes and image tolerance are shrinking rapidly compared
with l0/NA. Chapter 9 discusses significant causes of patterning nonuniformity arising
from optical imaging, and techniques for their measurement.

I am thankful to many friends and colleagues during the course of this project. In the
first place, I am grateful to Dr. Anthony Yen for encouraging me to write a text on
this topic. I am indebted to Dr. Timothy Brunner, Dr. Gregg Gallatin, Professor
Andrew Neureuther, Dr. Alan Rosenbluth, Dr. Frank Schellenberg, and Dr. Yen for their
comments and their meticulous review of the manuscript. I am much beholden to my
dissertation advisors, Professor Andrew Neureuther and Professor William Oldham, for
introducing me to microlithography and for their lessons of wisdom. It is an honor to
have the Foreword of this book written by Professor Neureuther.

I am obliged to Dr. Gallatin and Dr. Yen for their suggestions on development of the
Rayleigh-Sommerfeld diffraction formula in 3.5, and to Dr. Rosenbluth for his
exposition of the obliquity factor in 4.2. I would also like to acknowledge Dr.
Wilhelm Ulrich's permission for reproduction of the illustration in Fig. 2.6.
Publication of this book is the culmination of years of work by the SPIE Press
staff, to whom I owe much thanks.

I have many fond memories in writing this text, as my wife Aida and I often agonize
side by side on our respective writings. I hope the readers will also enjoy this book,
and privilege me with suggestions for improvement.